Find the interval in which the inner loop exists.
For this we take
So inner loop lies between the lines and
Since curve is symmetric about the vertical polar axis. So we canf ind the area of the loop by making double the area between and
Then Area of the loop is
The inner loop of the curve is bounded by the 2 points where r=0. Plug that in and you get sin? = -1/2, so ? = 210 and 330 degrees, or 7?/6 and 11?/6. So the integral needed is
Apply the Sum Rule:
Compute the boundaries: